IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009
3589
Ergodic Capacity Upper Bound for Dual MIMO Ricean Systems:
Simplified Derivation and Asymptotic Tightness
Michail Matthaiou, Member, IEEE, Yannis Kopsinis, Member, IEEE,
David I. Laurenson, Member, IEEE, and Akbar M. Sayeed, Senior Member, IEEE
Abstract—An analytical upper bound on the ergodic capacity
of Multiple-Input Multiple-Output (MIMO) systems is deduced
with the aid of a simplified approach that relies on a fundamental
power normalization. Given their high practical usability, we
are particularly interested in dual configurations where both
ends deploy two antenna elements. Contrary to the majority of
related studies, where only the common case of Rayleigh fading
is considered, our analysis is extended to account for Ricean
fading where a deterministic Line-of-Sight (LoS) component
exists in the communication link and both ends are affected
by spatial correlation. In the following, it is shown that the
proposed bound is applicable for any arbitrary Signal-to-Noise
Ratio (SNR) and rank of the mean channel matrix. In fact, we
consider both conventional and optimized MIMO configurations
with equal LoS eigenvalues. Moreover, the tightness of the bound
is explored where it is demonstrated that as the SNR tends to
zero the bound becomes asymptotically tight; at high SNRs, the
offset between empirical capacity and the bound is analytically
computed which implies that an explicit asymptotic capacity
expression can ultimately be obtained.
Index Terms—MIMO systems, ergodic capacity, Ricean fading,
spatial fading correlation.
I. I NTRODUCTION
O
VER the last years, a considerable amount of research
interest has been devoted to Multiple-Input MultipleOutput (MIMO) systems in response to the increasing demand
for higher data rates. The pioneering works of Foschini [1] and
Telatar [2] demonstrated the dramatic performance enhancement when multiple antenna elements are employed at both
ends of a radio link. In order to comprehend the advantages
of this promising technology, one of the most interesting
topics that needs to be addressed is the derivation of efficient
analytical capacity bounds. By doing so, it is anticipated
that the design of practical and simulated MIMO systems
would be greatly enriched. Please note that throughout the
paper our main interest will lie in dual MIMO configurations
which are likely to be used in many future practical systems
Paper approved by N. Jindal, the Editor for MIMO Techniques of the IEEE
Communications Society. Manuscript received August 1, 2008; revised April
4, 2009.
M. Matthaiou is with the Institute for Circuit Theory and Signal Processing,
Technische Universität München (TUM), Arcistrasse 21, 80333, Munich,
Germany (e-mail: [email protected]).
Y. Kopsinis and D. I. Laurenson are with the Institute for Digital Communications (IDCOM), Joint Research Institute for Signal and Image Processing,
The University of Edinburgh, EH9 3JL, Edinburgh, U.K. (e-mail: {Y.Kopsinis,
Dave.Laurenson}@ed.ac.uk).
A. M. Sayeed is with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706 USA (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TCOMM.2009.12.080393
(e.g. hand-held devices), thanks to their small size and low
implementation cost.
The majority of related studies documented in literature
consider the tractable case of Rayleigh fading where no
direct Line-of-Sight (LoS) exists in the radio channel and
the multipath richness is sufficiently high. In this case, a
plethora of results is available for various scenarios (see [1]–
[4] and references therein). In most real-life channels though,
the assumption of Rayleigh fading is often violated due to
either a specular wavefront or a strong direct component;
then, the entries of the channel matrix can be more effectively
modeled by the Ricean distribution. Surprisingly, significantly
fewer publications focusing on MIMO Ricean channels have
been reported. This trend can be attributed to the difficulty
in manipulating hypergeometric functions with two matrix
arguments of non-central Wishart matrices compared to the
one matrix argument of central (zero-mean) Wishart matrices
tied to Rayleigh fading.
The first analytical bounds on MIMO Ricean capacity can
be found in [5]–[7] where the assumption of uncorrelated
fading at both ends was adopted. In [8], [9], these results
were extended to account for spatial correlation at a single end
while the general case of double-sided spatial correlation was
addressed in [10]. The common characteristic of the above
cited papers though ([5]–[10]), is that they are limited to
the case of rank-1 LoS matrices. The derivation of capacity
bounds in the general case of arbitrary-rank mean matrices
has been separately addressed in [11]–[14]. In [11] and [12],
very tight lower and upper capacity bounds for semi-correlated
MIMO Ricean channels were respectively proposed. The work
in [13] represents so far the more general approach in the
area since it derives several lower and upper bounds assuming
all different types of spatial correlation. However, the general
upper bound is given as an infinite summation of Hayakawa
polynomials, which the authors acknowledge as quite involved
and computationally inefficient. On the other hand, the upper
bound deduced in [14], which relies on the theory of quadratic
forms, is far more tractable and hence will serve as our
reference bound hereafter. Whilst the authors in [14] focus
on the general properties of quadratic forms with respect
to MIMO capacity bounds, our primary goal is to provide
a simplified bound derivation and thereafter investigate the
implications of the model parameters (Ricean 𝐾-factor and
spatial correlation) on the bound performance.
More specifically, the contributions of the present paper
are threefold. Firstly, an alternative derivation of the upper
bound in [14] is proposed for the dual MIMO case under
c 2009 IEEE
0090-6778/09$25.00 ⃝
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009
consideration. Our analysis relies on a well-established power
normalization scheme which simplifies significantly all the
mathematical formulations. Secondly, using the same normalization as a starting point, the asymptotic tightness of the
upper bound is explored in the low and high Signal-to-Noise
Ratio (SNR) regimes. In fact, it is explicitly shown that as the
SNR tends to zero the bound becomes asymptotically tight,
whereas, at infinitely high SNRs the offset from the true capacity is analytically determined as well; hence, exact capacity
expressions can be derived in the corresponding asymptotic
cases. Finally, in order to formulate a broad framework, two
different classes of LoS MIMO systems are assessed, namely a
conventional low-rank (LR) and an optimized high-rank (HR)
configuration which yields two equal LoS eigenvalues and is
of high practical importance [15]-[17].
The paper is organized as follows: In Section II, some
essential definitions related to non-central Wishart matrices
and quadratic forms are given. In Section III, the underlying
MIMO Ricean channel model is presented along with the
statistics of the channel matrix. Section IV presents upper
capacity bounds for different categories of MIMO systems. In
Section V, we focus on the tightness of the proposed bound
while the numerical results are given in Section VI. Finally,
Section VII summarizes the key findings.
Notation: We use upper and lower case boldfaces to denote
matrices and vectors. The symbol ∼ 𝒞𝒩 𝑚,𝑛 (M, Σ) stands for
a 𝑚 × 𝑛 complex normally distributed matrix with mean M
and covariance Σ. The entries of a 𝑚 × 𝑛 matrix A are given
by {A}𝑖,𝑗 where 𝑖 = 1, . . . , 𝑚 and 𝑗 = 1, . . . , 𝑛. The symbols
𝑇
𝐻
−1
(⋅) , (⋅) and (⋅)
correspond to transposition, Hermitian
transposition and matrix inversion respectively whereas ⊗ is
the Kronecker product; we use det(⋅) and ∣ ⋅ ∣ to interchangeably denote the determinant while ∥⋅∥𝐹 returns the Frobenius
norm. Finally, etr(⋅) is a shorthand notation for exp(tr(⋅)).
II. C OMPLEX N ON -C ENTRAL W ISHART M ATRICES A ND
Q UADRATIC F ORMS
Definition 1: Let the 𝑚×𝑛 complex matrix X, with 𝑚 ≤ 𝑛,
be distributed according to X ∼ 𝒞𝒩 𝑚,𝑛 (M, Σ ⊗ I𝑛 ), where
Σ ∈ ℂ𝑚×𝑚 is a positive definite Hermitian matrix. Then,
the matrix S = XX𝐻 follows the complex non-central
Wishart distribution with 𝑛 degrees of freedom and noncentrality matrix Ω = Σ−1 MM𝐻 , commonly denoted as
S ∼ 𝒞𝒲 𝑚 (𝑛, Σ, Ω).
Definition 2: Let the 𝑚×𝑛 complex matrix X, with 𝑚 ≤ 𝑛,
be distributed according to X ∼ 𝒞𝒩 𝑚,𝑛 (M, Σ ⊗ Ψ), where
Σ ∈ ℂ𝑚×𝑚 and Ψ ∈ ℂ𝑛×𝑛 are positive definite Hermitian
matrices. Then, the matrix Q = XΛX𝐻 , with Λ ∈ ℂ𝑛×𝑛 ,
follows a non-central matrix-variate quadratic form commonly
denoted as Q ∼ 𝒞𝒬𝑚,𝑛 (Λ, Σ, Ψ, M).
Please note that non-central quadratic forms degenerate into
non-central Wishart matrices when Ψ = I𝑛 and when either
Λ = I𝑛 or Λ is idempotent with rank 𝐿 ≥ 𝑚 [13]. The
following theorem returns the 𝜐-th moment of the determinant
of 2 × 2 complex quadratic forms.
Theorem 1: Let Q ∼ 𝒞𝒬2,2 (I2 , Σ, Ψ, M). Then, the 𝜐-th
moment of its determinant ∣Q∣ is given by
[
]
𝜐
𝜐 Γ̃2 (𝜐 + 2)
˜
(1)
𝐸 [∣Q∣ ] = ∣ΣΨ∣
1 𝐹1 (−𝜐; 2; −Θ)
Γ̃2 (2)
∏
where Γ̃𝑚 (𝑛) = 𝜋 𝑚(𝑚−1)/2 𝑚
𝑗=1 Γ(𝑛− 𝑗 + 1) is the complex
multivariate gamma function, with Γ(⋅) being the standard
gamma function, while 1 𝐹˜1 (𝑎, 𝑏, A) is the hypergeometric
function of one matrix argument; the involved matrices are
accordingly Θ = Ψ−1 M̄M̄𝐻 and M̄ = Σ−1/2 M.
Proof: A detailed proof is given in Appendix A.
It should be noted that the theorem is applicable only to
2 × 2 quadratic forms since for matrix
of 𝑚 × 𝑛, a
)
( 𝑛sizes
subsets needs to
finite summation over a collection of 𝑚
take place [14]. A simplified formula can now be obtained for
the first-order moment of the determinant.
Corollary 1: For 𝜐 = 1, (1) reduces to
(
)
1
1
𝐸 [∣Q∣] = 2∣ΣΨ∣ 1 + tr(Θ) + det (Θ) .
(2)
2
2
Proof: The authors in [13] showed that for any square
matrix B ∈ ℂ𝑚×𝑚 , its hypergeometric function 1 𝐹˜1 (𝑐; 𝑑; B)
can be expressed according to
(
)
det 1 𝐹˜1 (𝑐 − 𝑚 + 𝑗; 𝑑 − 𝑚 + 𝑗; 𝑏𝑖 ) 𝑏𝑗−1
𝑖
˜
1 𝐹1 (𝑐; 𝑑; B) =
𝑚
∏
(𝑏𝑗 − 𝑏𝑖 )
𝑖<𝑗
(3)
where 𝑏1 , 𝑏2 , . . . , 𝑏𝑚 is the set of non-zero eigenvalues of B.
After taking into account the following properties of the scalar
hypergeometric functions
1
1 − 2𝑧 + 𝑧 2
(4)
2
1 2
˜
(5)
1 𝐹1 (−1, 2; 𝑧) = 1 − 𝑧
2
it is trivial to show that for the dual case under consideration
1
˜
(2 + 𝜃1 + 𝜃2 + 𝜃1 𝜃2 )
(6)
1 𝐹1 (−1; 2; −Θ) =
2
where 𝜃1 , 𝜃2 are the eigenvalues of Θ. The proof concludes
after recalling that the sum and product of the eigenvalues
return the trace and the determinant of a matrix.
˜
1 𝐹1 (−2, 1; 𝑧)
=
III. MIMO C HANNEL M ODEL
As was previously highlighted, we are particularly interested in dual MIMO configurations where both the Tx and Rx
are affected by spatial correlation. Under Ricean fading conditions, the channel transfer function matrix H consists of a
spatially deterministic specular component HL and a randomly
distributed component HW which accounts for the scattered
signals. Then, the underlying channel model becomes
√
√
𝐾
1
1/2
HL +
R1/2 HW R𝑡
H=
(7)
𝐾 +1
𝐾 +1 𝑟
where 𝐾 is the Ricean 𝐾-factor expressing the ratio of powers
of the free-space signal and the scattered waves. The receive
and transmit correlation matrices are respectively defined as
{(
{
}
)𝑇 }
.
(8)
R𝑟 ≜ 𝐸H HH𝐻 and R𝑡 ≜ 𝐸H H𝐻 H
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MATTHAIOU et al.: ERGODIC CAPACITY UPPER BOUND FOR DUAL MIMO RICEAN SYSTEMS: SIMPLIFIED DERIVATION AND ASYMPTOTIC . . .
z
zǯ
wavefronts in the near-field. Then, the channel matrix becomes
full-rank and delivers two equal LoS eigenvalues. For the
assumed model, the optimum inter-element spacing is [17]
√
𝜆𝐷
.
(11)
𝑠1 = 𝑠2 = 𝑠𝑜𝑝𝑡 ≈
2 cos2 𝜃
[-s1sinș, 0, s1cosș]
s1
ș
y
[0,0,0)]
3591
x
ș
[D-s2sinș, 0, s2cosș]
D
s2
[D, 0, 0]
xǯ
Fig. 1. A general architecture of a 2 × 2 MIMO system with ULAs at
both ends (side view). The coordinates of all elements with regard to the new
coordinate system 𝑥′ 𝑦𝑧 ′ are also included. The axis rotation by an angle 𝜃
arranges the array origins to lie on the same axis.
The entries of HW are commonly modeled as i.i.d. zeromean, unit variance complex Gaussian random variables and,
consequently, the channel matrix is distributed according to
(√
)
𝐾/(𝐾 + 1)HL , (𝐾 + 1)−1 R𝑟 ⊗ R𝑡 . (9)
H ∼ 𝒞𝒩 2,2
Throughout the paper, our purpose is to compare the MIMO
gains of different configurations and therefore the capacities
should be analyzed independently of the average SNR. This
is achieved by normalizing the channel matrices so that
following condition is fulfilled
)]
[ (
[
]
𝐸 ∥H∥2𝐹 = 𝐸 tr HH𝐻 = 4.
(10)
We recall that the key normalization in (10) has been widely
adopted into the capacity characterization of MIMO systems [1], [2], [12], [16], [17]. From a physical viewpoint, the
path-loss effects are removed and the system is assumed to
have perfect power control.
In Fig. 1, a side view of the considered MIMO system
is depicted where both ends are equipped with two-element
Uniform Linear Arrays (ULAs) and the distance between the
first element of each array is 𝐷. The inter-element spacings
are 𝑠1 (Tx) and 𝑠2 (Rx).
In the following, we will explore two different LoS configurations, namely a LR and an optimized HR configuration.
The former represents any conventional architecture with interelement spacings of the order of wavelength 𝜆, and arrays
located in each other’s far-field; this means that the LoS
signals are basically plane wavefronts whose rays’ phases are
inherently correlated to a great extent. These architectures are
normally rank-deficient and offer a minimal spatial multiplexing gain since the transmitted LoS signals impinging on the Rx
carry almost identical spatial characteristics and therefore their
differentiation becomes laborious. The latter configuration
belongs to a family of specifically designed full-rank LoS
configurations which aim to assign a phase difference of 𝜋/2
between the LoS wavefronts [15]–[17]. This is achieved by
appropriately positioning the antenna elements at both ends
of the link, so that the LoS signals propagate as spherical
We emphasize that for optimized configurations the effects
of spatial correlation are rather weak due to the increased antenna spacing. However, in the following analysis the presence
of correlation is considered, for the fairness of comparison
with conventional configurations.
IV. E RGODIC C APACITY U PPER B OUNDS
In this section, an alternative simplified derivation for the
upper bound of dual MIMO configurations (as given in [14]) is
presented based on the power normalization of (10). Assuming
that the Rx has perfect CSI while the Tx knows neither the
statistics nor the instantaneous CSI, a sensible choice for the
Tx is to split the total amount of power equally amongst all
data streams. Then, the ergodic MIMO capacity for an SNR
per receiver branch, 𝜌, is given by the following well-known
formula [1], [2]
[
(
(
))]
𝜌
𝐶 = 𝐸H log2 det I2 + HH𝐻
(bits/s/Hz). (12)
2
A. Double-Sided Correlated Ricean and Rayleigh Fading
In the general case of double-sided correlated Ricean fading,
where the channel matrix is distributed according to (9), the
instantaneous MIMO correlation matrix W̃ = HH𝐻 exhibits
the following non-central quadratic distribution
(
)
√
W̃ ∼ 𝒞𝒬2,2 I2 , R𝑟 /(𝐾 + 1), R𝑡 , 𝐾/(𝐾 + 1)HL .
(13)
Theorem 2: The ergodic capacity of a 2 × 2 doublesided
correlated MIMO Ricean channel with mean matrix
√
𝐾/(𝐾 + 1)HL , receive correlation matrix (𝐾 + 1)−1 R𝑟 ,
and transmit correlation matrix R𝑡 , is analytically upper
bounded by
(
)
𝛾𝜌2
𝐶 ≤ log2 1 + 2𝜌 +
(14)
2
where the parameter 𝛾 and the matrix Θ are respectively
)
(
1
1
(15)
𝛾 = (𝐾 + 1)−2 ∣R𝑟 ∣∣R𝑡 ∣ 1 + tr(Θ) + det (Θ)
2
2
(
)𝐻
𝐻
−1/2
−1/2
Θ = 𝐾R−1
R
R
H
H
. (16)
L
L
𝑡
𝑟
𝑟
Proof: An alternative way to express the ergodic MIMO
capacity is through the real positive eigenvalues 𝑤
˜1 , 𝑤
˜2 of W̃.
Then, (12) can be rewritten as
(
[
𝜌 )(
𝜌 )]
(17)
˜1 1 + 𝑤
˜2 .
𝐶 = 𝐸 log2 1 + 𝑤
2
2
Expanding (17), we can easily get
[
(
( ))]
𝜌2
𝜌
˜1 + 𝑤
det W̃
𝐶 = 𝐸 log2 1 + (𝑤
˜2 ) +
.
2
4
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(18)
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009
Taking into account that log(⋅) is a concave function and
making use of the Jensen’s inequality we obtain
(
( )])
𝜌2 [
𝐶 ≤ log2 1 + 2𝜌 + 𝐸 det W̃
.
(19)
4
The last inequality
follows from the power normalization in
[ ( )]
˜2 ] = 4. The upper bound
(10), i.e. 𝐸 tr W̃ = 𝐸 [𝑤
˜1 + 𝑤
in (14) follows immediately after introducing Corollary 1 and
simplifying.
It is worth mentioning that the same upper capacity bound
for 2 × 2 MIMO systems was originally given in [14].
More specifically, by setting 𝑛 = 2 in [14, Eq. (36)] and
evaluating the first three basic elementary functions and minor
determinant representations involved, the same result can be
obtained after some algebraic manipulations. In the present
case though, a novel derivation is introduced, through the
power normalization of (10), which simplifies extensively the
overall analysis. Please note that the proposed derivation is tied
to dual configurations since for larger MIMO setups we end up
with a series of eigenvalue cross-products that are created after
expanding (12). By inspection of (14), it can be inferred that
the proposed upper bound is rather simple as it just requires
the computation of the elementary functions of three different
deterministic matrices (R𝑟 , R𝑡 and Θ) only once.
In the case of double-sided correlated Rayleigh fading
(𝐾 = 0), the channel matrix is distributed according to
H ∼ 𝒞𝒩 2,2 (02 , R𝑟 ⊗ R𝑡 ) and the upper bound in (14)
reduces to
)
(
𝜌2
𝐶 ≤ log2 1 + 2𝜌 + ∣R𝑟 R𝑡 ∣
(20)
2
which is in perfect line with the bound derived in [4, Eq. (25)].
B. Uncorrelated Ricean and Rayleigh Fading
Due to space constraints, the case of single-side correlation
is omitted herein since the formulations are based on exactly
the same concept as before. We now consider the special
case of both ends exhibiting uncorrelated i.i.d. Ricean fading.
Under these circumstances, the channel matrix is distributed
as
(√
)
𝐾
1
HL ,
I2 ⊗ I2 .
(21)
H ∼ 𝒞𝒩 2,2
𝐾 +1
𝐾 +1
Corollary 2: The ergodic capacity of a 2√
× 2 uncorrelated
MIMO Ricean channel with mean matrix 𝐾/(𝐾 + 1)HL
and receive correlation matrix R𝑟 = (𝐾 + 1)−1 I2 is analytically upper bounded by
(
)
𝛽𝜌2
𝐶 ≤ log2 1 + 2𝜌 +
(22)
2
)
(
where 𝛽 = 1 + 2𝐾 + 0.5𝐾 2 det (T) /(𝐾 + 1)2 and T =
HL H𝐻
L .
Proof: This corollary is a direct consequence of (14) after
taking into account that Θ ≡ Ω ≡ 𝐾T for the case of
uncorrelated fading at both ends. Furthermore, it holds that
tr (𝐾T) = 𝐾tr (T) and det (𝐾T) = 𝐾 2 det (T) and given
that the entries of the deterministic LoS component matrix
are normally modeled as unit-amplitude complex exponentials [15]–[17], it is trivial to show that tr (T) = 4.
The case of i.i.d. Rayleigh fading is obtained directly from
(7), by setting 𝐾 = 0 and R𝑟 = R𝑡 = I2 . The upper bound
in (22) then reduces to (𝛽 = 1)
(
)
𝜌2
𝐶 ≤ log2 1 + 2𝜌 +
(23)
2
which is identical with the results presented in [3, Theorem
2], [4, Eq. (22)] and [12, Eq. (5)].
V. T IGHTNESS OF THE U PPER B OUND
In this section, the asymptotic tightness of the proposed
upper bound is investigated in low and high-SNR regimes. To
this end, we define the absolute error, 𝜖, inserted by an upper
bound, 𝑈 , as 𝜖 = 𝑈 − 𝐶.
Corollary 3: The upper bound in (14) becomes asymptotically tight as the SNR tends to zero.
Proof: We begin with further upper bounding the ergodic
capacity (14) recalling that ln(1 + 𝑥) ≤ 𝑥
(
)
1
𝛾𝜌2
𝐶≤
2𝜌 +
.
(24)
ln 2
2
Following [18, Eq. (23)], we can lower bound the ergodic
capacity as
(
)]
[
𝜌
2
𝐶 ≥ 𝐸 log2 1 + ∥H∥𝐹
2
]
]
[
1 ( 𝜌 )2 [
𝜌
2
𝐸 ∥H∥𝐹 −
𝐸 ∥H∥4𝐹 .(25)
≥
2 ln 2
2 ln 2 2
The second line follows from the property ln(1+𝑥) ≥ 𝑥− 12 𝑥2 .
We can now subtract (25) from (24) and then the absolute error
𝜖 of the proposed upper bound becomes
(
])
1 [
𝜌2
4
𝛾 + 𝐸 ∥H∥𝐹
(26)
𝜖=
2 ln 2
4
which asymptotically tends to zero as 𝜌 → 0.
Corollary 4: As the SNR 𝜌 → ∞, the absolute error
inserted by the upper bound in (14) tends to
(
( ))]
[
.
(27)
𝜖 = log2 (2𝛾) − 𝐸 log2 det W̃
Proof: As 𝜌 → ∞, the upper bound 𝑈 in (14), becomes
(𝜌)
𝑈 ≈ log2 (2𝛾) + 2 log2
.
(28)
2
In (18), the quadratic term becomes significantly larger at high
SNR and thus the ergodic capacity may be approximated as
[
( 2
( ))]
𝜌
det W̃
𝐶 ≈ 𝐸 log2
4
(𝜌)
(
( ))]
[
.
(29)
+ 𝐸 log2 det W̃
= 2 log2
2
Subtracting (29) from (28), yields (27).
From the above equation, it is apparent that the bound’s
error is given in a non-analytical form; in this light, the crucial
issue is to determine the expectation of the logdet function
of a complex non-central quadratic matrix, which involves a
nonlinear log function.
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MATTHAIOU et al.: ERGODIC CAPACITY UPPER BOUND FOR DUAL MIMO RICEAN SYSTEMS: SIMPLIFIED DERIVATION AND ASYMPTOTIC . . .
where 𝜃1 , 𝜃2 are the eigenvalues of Θ, given in (16), while
the digamma functions 𝜓(𝑥) are defined as
𝜓(𝑥) ≜
Γ′ (𝑥)
𝑑
ln Γ(𝑥) =
.
𝑑𝑥
Γ(𝑥)
(31)
The polynomial terms Λ1 (Θ) and Λ2 (Θ) are essentially
polynomial functions of the eigenvalues 𝜃1 and 𝜃2 , or
Λ1 (Θ)
= 𝜃2 ℎ1 (𝜃1 ) − ℎ2 (𝜃1 )
(32)
Λ2 (Θ)
= ℎ2 (𝜃2 ) − 𝜃1 ℎ1 (𝜃2 )
(33)
where
ℎ1 (𝑥) =
∞
∑
𝒫(𝑘, 𝑥)
𝑘=1
𝑘
, ℎ2 (𝑥) = 𝑥
∞
∑
𝒫(𝑘, 𝑥)
𝑘=1
𝑘+1
(34)
with 𝒫(𝑎, 𝑥) being the regularized gamma function [19, Eq.
(6.5.1)].
Proof: A detailed proof is given in Appendix B.
Please note that the derivation of the formula (30) relies,
without loss of generality, on the assumption of two non-zero
eigenvalues of the matrix Θ. For the case of rank-1 matrices,
a similar analysis can be followed and the interested readers
are referred to [13] for a thorough discussion. Evidently, after
replacing (30) into (27), we can obtain an analytical formula
for the bound’s offset at high SNRs, for the general case of
double-sided correlated Ricean fading. More importantly, this
result can be further used to deduce exact capacity expressions
in the high-SNR regime.
When the channel exhibits i.i.d. Ricean fading and both
ends are employed with optimally designed arrays, the LoS
component yields two equal eigenvalues and the following
corollary is introduced.
Corollary 5: As the SNR 𝜌 → ∞, the absolute error
inserted by the upper bound for the case of i.i.d. Ricean fading
and optimized LoS configurations tends to
[
1
𝜓(1) + 𝜓(2) − 2 ln(𝐾 + 1)
𝜖 = log2 (2𝛽) −
ln 2
]
∞
∑
(2𝑘 + 1)𝛾(𝑘, 𝜔) − 𝜔𝑒−𝜔 𝜔 𝑘−1
+
(35)
(𝑘 + 1)!
𝑘=1
where 𝜔 = 𝜔1 = 𝜔2 represents any∫ of the two equal
𝑥
eigenvalues of Ω ≡ 𝐾T and 𝛾(𝑎, 𝑥) = 0 𝑡𝑎−1 𝑒−𝑡 𝑑𝑡 is the
lower incomplete gamma function.
Proof: The proof starts by noting that for i.i.d. Ricean
fading Θ should be replaced by Ω ≡ 𝐾T in all manipulations.
Further, in the specific case of optimized configurations, the
equality of eigenvalues leads to a division by zero in (30). In
order to circumvent this singularity, we employ de l’Hôpital’s
35
Upper bound
Simulated ergodic capacity
30
Ergodic capacity, (bits/s/Hz)
Theorem 3:( Let us assume that√the considered) matrix
W̃ ∼ 𝒞𝒬2,2 I2 , R𝑟 /(𝐾 + 1), R𝑡 , 𝐾/(𝐾 + 1)HL . Then
the first-order moment of the logarithm of its determinant is
[
[
(
( ))]
1
𝐸 log2 det W̃
=
𝜓(1) + 𝜓(2) − 2 ln(𝐾 + 1)
ln 2
]
1
(Λ1 (Θ) + Λ2 (Θ))
(30)
+ ln ∣R𝑟 R𝑡 ∣ −
𝜃1 − 𝜃2
3593
25
20
K = 10 dB
15
K = 0 dB
10
5
0
0
10
20
30
40
50
SNR, [dB]
Fig. 2. Upper bound and simulated ergodic capacity as a function of the
SNR for an optimized LoS configuration (𝛿𝑅 = 0.2 and 𝛿𝑇 = 0.5).
rule to get a solution at the limit 𝜔1 → 𝜔2 . In particular, the
last term in (30) can be rewritten after some algebra as
]
[
𝑑
(𝜔ℎ1 (𝜔 + 𝜀) − ℎ2 (𝜔 + 𝜀)) − ℎ1 (𝜔).
(36)
lim
𝜀→0 𝑑𝜀
Taking into account that
𝑑
𝑒−𝑥 𝑥𝑎−1
𝒫(𝑎, 𝑥) =
𝑑𝑥
Γ(𝑎)
(37)
and after some algebraic manipulations we end up with
𝑑
lim ℎ1 (𝜔 + 𝜀) =
𝜀→0 𝑑𝜀
𝑑
ℎ2 (𝜔 + 𝜀) =
𝜀→0 𝑑𝜀
lim
∞
∑
𝑒−𝜔 𝜔 𝑘−1
𝑘=1
∞
∑
𝑘=1
𝑘Γ(𝑘)
(38)
ℎ2 (𝜔)
𝑒−𝜔 𝜔 𝑘−1
+
. (39)
(𝑘 + 1)Γ(𝑘)
𝜔
Substituting (38)–(39) into (36), factorizing and simplifying
yields (35).
VI. N UMERICAL R ESULTS
In this section, the theoretical analysis presented in Sections IV and V is validated through a set of Monte-Carlo simulations. Assuming a carrier frequency of 5.9 GHz, 𝐷 = 5.3852
m and 𝜃 = 21.80∘ the optimum inter-element spacings via (11)
are 𝑠1 = 𝑠2 = 39.85 cm (7.84𝜆) whereas for the conventional
configuration the spacings are taken 𝑠1 = 𝑠2 = 2.54 cm
(0.5𝜆). Then, the associated sets of the LoS eigenvalues of
−5
T = HL H𝐻
L are (2, 2) and (3.999, 4.08 × 10 ), respectively.
Throughout the simulations, we adopt the widely used exponential correlation model thanks to its inherent simplicity.
Then, the entries of R𝑟 and R𝑡 in (7) can be modeled
as {R𝑟 }𝑖,𝑗 = (𝛿𝑅 )∣𝑖−𝑗∣ and {R𝑡 }𝑖,𝑗 = (𝛿𝑇 )∣𝑖−𝑗∣ , where
𝛿𝑅 , 𝛿𝑇 ∈ [0, 1).
After generating 50,000 Monte-Carlo realizations of the
channel matrix according to (7) and setting 𝛿𝑅 = 0.2 and
𝛿𝑇 = 0.5, the proposed bound in (14) is firstly evaluated
against the SNR for different values of the 𝐾-factor.
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3594
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009
35
14
Upper bound
Simulated ergodic capacity
13
Ergodic capacity, (bits/s/Hz)
Ergodic capacity, (bits/s/Hz)
30
25
20
K = 0 dB
15
K = 10 dB
10
5
0
optimized
12
11
10
conventional
9
Upper bound
Simulated ergodic capacity
8
0
10
20
30
40
7
50
0
5
10
SNR, [dB]
From Fig. 2 and 3, it can been seen that the bound is
remarkably tight for the optimized configuration and, likewise,
performs satisfactorily for conventional configurations. As
anticipated, in the low-SNR regime both bounds converge
asymptotically to the empirical values of ergodic capacity.
Generally speaking, the bound becomes tighter as 𝐾 increases
and SNR decreases which is in agreement with the conclusions
of [9], [10], [12]. More importantly, ergodic capacity benefits
from the presence of strong non-fading components when both
ends are equipped with specifically designed arrays. This is
a result of the two orthogonal LoS MIMO subchannels that
are established and contradicts the common belief that high
𝐾-factors limit the advantages of MIMO technology [6], [7],
[9].
In Fig. 4, the effects of the 𝐾-factor on the performance of
the proposed bound are investigated, where it is again apparent
that both bounds become tighter with an increasing 𝐾-factor.
For Rayleigh-fading conditions though, or 𝐾 ≤ 0 dB, the
achieved tightness is degraded and, under these circumstances,
it is sensible to use more efficient bounds which are inherently
tied to Rayleigh channels (see for instance [3], [4]). Once
more, the superiority of the optimized configurations is illustrated as 𝐾 gets larger whereas in the limit, 𝐾 → ∞ dB,
conventional configurations degenerate into a single-path link.
On the other hand, for 𝐾 ≤ 0 dB the advantages of optimized
configurations diminish and in the limit, 𝐾 → −∞ dB, the
LoS component vanishes and we end up with a pure i.i.d.
Rayleigh channel.
In Fig. 5, the relationship between spatial correlation and
MIMO capacity is addressed. The effects of the former
become less significant (smaller dynamic range) as the 𝐾factor gets higher for both configurations, i.e. high 𝐾-factors
provide robustness against spatial correlation. As expected,
the optimized setup with large inter-element spacings remains
unaffected by the level of correlation; hence, it offers almost
the same ergodic capacity regardless of the values of 𝛿𝑅 and
𝛿𝑇 . The conventional configuration, however, suffers from
spatial correlation with the ergodic capacity decreasing as
20
Fig. 4. Upper bound and simulate ergodic capacity as a function of the
𝐾-factor (𝛿𝑅 = 0.2, 𝛿𝑇 = 0.5 and 𝜌 = 20 dB).
14
K = 10 dB
13
Ergodic capacity, (bits/s/Hz)
Fig. 3. Upper bound and simulated ergodic capacity as a function of the
SNR for a conventional LoS configuration (𝛿𝑅 = 0.2 and 𝛿𝑇 = 0.5).
15
K−factor, [dB]
K = 0 dB
12
11
K = 0 dB
10
9
Upper bound−opt
Simulated ergodic capacity−opt
Upper bound−conv
Simulated ergodic capacity−conv
8
7
0
0.1
0.2
0.3
0.4
0.5
K = 10 dB
0.6
0.7
0.8
Correlation coefficient
Fig. 5. Upper bound and simulated ergodic capacity as a function of the
correlation coefficients 𝛿𝑅 = 𝛿𝑇 (𝜌 = 20 dB).
correlation gets higher while the tightness of the corresponding
bound is relatively improved in the high-correlation regime.
This outcome is in agreement with [4], [10], [12].
At the last stage of the evaluation process, we consider
the high-SNR deviation between the ergodic capacity and
the proposed upper bound using the closed-form formulae
in (27), (30) and (35). In Fig. 6, these analytical curves are
overlaid with the outputs of a Monte-Carlo simulator with
the match being remarkably good. The error associated with
optimized configurations is systematically lower than that of
conventional ones, as a result of the rank-deficiency of the
former. What is more, it appears that the latter error has a much
smaller dynamic range revealing that a high 𝐾-factor does
not have an extensive impact on its value. On the other hand,
the bound for the optimized configuration yields an enhanced
tightness as 𝐾 increases and under strong Ricean conditions
the corresponding offset is minimized.
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MATTHAIOU et al.: ERGODIC CAPACITY UPPER BOUND FOR DUAL MIMO RICEAN SYSTEMS: SIMPLIFIED DERIVATION AND ASYMPTOTIC . . .
1.4
Theorem 1]
[
]
[
) ]
(
𝜐 Γ̃2 (𝜐 + 2)
𝐻 𝜐
= det (Ψ)
𝐸 det X̄X̄
Γ̃2 (2)
× etr(−Θ)1 𝐹˜1 (𝜐 + 2; 2; Θ)
Analytical
1.2
Simulation
conventional
Absolute error
1
(41)
where we have made use of the basic property, det (CD) =
det (DC), for the determinant of the product of square matrices. The proof concludes after introducing the well-known
Kummer relation for hypergeometric functions of one matrix
argument, i.e. 1 𝐹˜1 (𝑎; 𝑏; S) = etr(S)1 𝐹˜1 (𝑏 − 𝑎; 𝑏; −S).
0.8
0.6
optimized
0.4
0.2
0
3595
A PPENDIX B
P ROOF OF T HEOREM 3
0
5
10
15
20
K−factor, [dB]
Fig. 6. Analytical and simulated absolute error of the upper bound in the
high-SNR regime as a function of the 𝐾-factor (𝛿𝑅 = 0.2 and 𝛿𝑇 = 0.5).
VII. C ONCLUSION
An alternative simplified derivation of a tractable upper
bound for the ergodic capacity of dual MIMO Ricean systems
has been proposed, with the key concept originating from
a widely used power normalization. The proposed bound
depends merely on the SNR and the expected value of the
determinant of either a non-central quadratic form or a noncentral Wishart matrix and, further, is not confined to the
common case of rank-1 LoS components. The tightness of
the bound was assessed in the low and high-SNR regions; in
the former, the bound becomes asymptotically tight whereas
in the latter the bound’s offset tends to a constant value that
was analytically determined and validated through MonteCarlo simulations. For the sake of completeness, we explored
two different classes of LoS configurations, these were a
conventional and an optimized architecture which benefits
from the presence of strong deterministic components by
offering two equal LoS eigenvalues. It was demonstrated that
the bound is remarkably tight for the optimized case and
marginally looser for conventional setups.
The proof begins with the following key transformation
𝐸 [ln 𝑥] =
𝑑
𝐸 [𝑥𝜐 ] ∣𝜐=0
𝑑𝜐
which holds since, by definition, 𝑥𝜐 = 𝑒𝜐 ln 𝑥 . By combining
(42) and (1) and denoting
(
( ))]
[ (
( ))]
[
1
=
𝐸 ln det W̃
𝜁 = 𝐸 log2 det W̃
(43)
ln 2
and
[
]
−1
𝜐 Γ̃2 (𝜐 + 2)
˜
𝜙(𝑣) = ∣(𝐾 + 1) R𝑟 R𝑡 ∣
1 𝐹1 (−𝜐; 2; −Θ)
Γ̃2 (2)
we can directly get
1 𝑑
1 𝑑
=
𝜙(𝑣)
ln 𝜙(𝑣)
𝜁=
ln 2 𝑑𝜐
ln 2 𝑑𝜐
𝜐=0
𝜐=0
We begin by expressing the determinant of the quadratic
form Q ∼ 𝒞𝒬2,2 (I2 , Σ, Ψ, M) as
[
)𝜐 ]
(
𝜐
𝐸 [det (Q) ] = 𝐸 det XX𝐻
)𝜐 ]
[
(
= 𝐸 det Σ1/2 X̄X̄𝐻 Σ1/2
[
)𝜐 ]
(
(40)
= det (Σ)𝜐 𝐸 det X̄X̄𝐻
where X̄ ∼ 𝒞𝒩 2,2 (M̄, I2 ⊗ Ψ). Using a result from [14]
with the aid of the Cauchy-Binet formula, it can be shown
that the matrix X̄𝐻 X̄ ∼ 𝒞𝒲 2 (2, Ψ, Θ). The expectation of
the determinant in (40) can now be evaluated through [13,
(44)
where the second equality follows since 𝜙(0) = 1. It is
easily seen that the above differentiation consists of three
multiplicative terms. Treating each one separately due to the
chain rule, the first term results in ln ∣R𝑟 R𝑡 ∣ − 2 ln(𝐾 + 1).
The second one, may be rearranged to give
)}
{ (
)}
Γ̃2 (𝜐 + 2)
𝑑 { (
𝑑
ln
ln Γ̃2 (𝜐 + 2) =
𝑑𝜐
𝑑𝜐
Γ̃2 (2)
𝜐=0
𝜐=0
{ 2
}
𝑑 ∑
=
ln (Γ(𝜐 + 2 − 𝑖 + 1)) (45)
𝑑𝜐
𝑖=1
A PPENDIX A
P ROOF OF T HEOREM 1
(42)
𝜐=0
which, through (31), yields 𝜓(1) + 𝜓(2). Focusing now on
the last term, we get (46) at the top of next page. The
proof concludes after introducing a useful result from [13,
Appendix II-B] for the nominator in (46) and simplifying the
denominator using 1 𝐹˜1 (𝛼; 𝛼; 𝑥) = exp(𝑥). Substituting (45)
and (46) into (44) and collecting common terms yields (30).
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3596
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 12, DECEMBER 2009
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